发表机构
Shanxi University; Hangzhou Normal University（山西大学; 杭州师范大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究 Virasoro 共形代数上的广义共形模，给出秩一自由模的完全分类，构造有限秩不可约模并建立不可约性判据，同时分类无限挠广义共形模。
AI 中文摘要
本文研究了 Virasoro 共形代数 $Vir$ 上的广义共形模，这一概念最初由 V. Kac 定义，但在文献中尚未被充分探索。首先，我们提出了一种扭曲构造，该构造可从任意李共形代数上已有的共形模生成新的广义共形模。然后，我们给出了 $Vir$ 上在 $\mathbb C[\partial]$ 上秩为一的自由非平凡广义共形模的完全分类，这进而导出了直线上的向量场李代数 $W_1=\bigoplus_{i\geq -1}\C L_i$ 上在 $\mathbb C[L_{-1}]$ 上秩为一的自由模的分类，其显式作用以 Bell 多项式表示。对于任意有限秩 $n$，我们构造了一类广义共形模 $V_{a,b,C(\partial)}$，并建立了以作用在 $\mathbb{C}(\partial)^n$ 上的微分算子 $\mathcal L=\frac{d}{d\partial}+C(\partial)$ 表示的完全不可约性判据。在 $n=2$ 的情形下，该判据归结为 Riccati 方程不存在有理解。随后，我们证明了任意有限秩的不可约广义共形模的存在性，这与共形情形形成了鲜明对比。我们还构造并分类了 $Vir$ 上的一族无限挠广义共形模，在适当条件下它们是非平凡且不可约的。
英文摘要
This paper investigates generalized conformal modules over the Virasoro conformal algebra $Vir$, a notion originally defined by V.~Kac but largely unexplored in the literature. First, we present a twisted construction that produces new generalized conformal modules over any Lie conformal algebra from existing conformal modules. Then we give a complete classification of non-trivial generalized conformal modules over $Vir$ that are free of rank one over $\mathbb C[\partial]$, which in turn yields a classification of modules over the Lie algebra \(W_1=\bigoplus_{i\geq -1}\C L_i\) of vector fields on a line that are free of rank one over $\mathbb C[L_{-1}]$ with explicit actions expressed in terms of Bell polynomials. For arbitrary finite rank $n$, we construct a class of generalized conformal modules $V_{a,b,C(\partial)}$ and establish a complete irreducibility criterion in terms of the differential operator $\mathcal L=\frac{d}{d\partial}+C(\partial)$ acting on $\mathbb{C}(\partial)^n$. In the case of $n=2$, this criterion reduces to the absence of rational solutions of a Riccati equation. Then we prove the existence of irreducible generalized conformal modules of arbitrary finite rank, revealing a striking contrast with the conformal setting. We also construct and classify a family of infinite torsion generalized conformal modules over $Vir$, which are non-trivial and irreducible under suitable conditions.
Comments32 pages
